With the evolution results in the previous section, we prove Theorem REF . The short time existence of \(\mathcal {C}_{t}\)  is from the short time existence of solution of the quasilinear parabolic equation for closed initial data. It's well known that if the mean curvature flow \(F_{t}\)  of the hypersurface in Euclidean space exists for only finite time interval \([0,q)\) , \(max_{F_{t}}|A|\rightarrow \infty \)  when \(t\)  approach to \(q\)  ([1]}). For the submanifold in Euclidean space with high codimension, such kind of lemma is proved by Andrews-Baker ([2]}). For the curve shortening flow's case, we give the proof of the following lemma only for the sake of completeness.
